1. Field of the Invention
The present invention relates to numerical modeling techniques used in engineering, and particularly to a computerized method for identifying Hammerstein models in which the linear dynamic part is modeled by a space-state model and the static nonlinear part is modeled using a radial basis function neural network (RBFNN).
2. Description of the Related Art
Many practical or “real world” systems inherently have nonlinear characteristics, such as saturation, dead-zones, etc. Identification of such systems is extremely important for prediction or control design. Nonlinear system identification has therefore been a subject of considerable research in the recent. One promising approach for nonlinear system identification is based on the assumption that the identified systems consist of relatively simple subsystems, and that the structure of the subsystems are known. This is commonly known as a “block-oriented approach”. Based on this approach, systems can be broken down into linear and nonlinear parts, separate from each other. The subsystems are then identified on the basis of the input-output signals of the whole system, and the a priori information about the system.
Block-oriented models have been applied to represent physical and biological systems. They have been used to model an electrical generator, communication bandpass circuits, a distillation column, the visual cortex, and pH control systems, along with an electrical drive. The Hammerstein Model belongs to a family of block-oriented models, and is made up of a memoryless nonlinear part followed by a linear dynamic part. It has been known to effectively represent and approximate several industrial processes, such as pH neutralization processes, distillation column processes, and heat exchange processes. Hammerstein models have also been used to successfully model nonlinear filters, biological systems, water heaters, and electrical drives.
A significant amount of research has been carried out on identification of Hammerstein models. Systems can be modeled by employing either nonparametric or parametric models. Nonparametric representations involve kernel regression or expansion of series, such as the Volterra series. This results in a theoretically infinite number of model parameters, and is therefore represented in terms of curves, such as step responses or bode diagrams. Parametric representations, such as state-space models, are more compact, as they have fewer parameters and the nonlinearity is expressed as a linear combination of finite and known functions.
Development of nonlinear models is the critical step in the application of nonlinear model-based control strategies. Nonlinear behavior is prominent in the dynamic behavior of physical systems. Most physical devices have nonlinear characteristics outside a limited linear range. In most chemical processes, for example, understanding the nonlinear characteristics is important for designing controllers that regulate the process. It is rather difficult, yet necessary, to select a reasonable structure for the nonlinear model to capture the process nonlinearities. The nonlinear model used for control purposes should be as simple as possible, warranting minimal computational load and, at the same time, retaining most of the nonlinear dynamic characteristics of the system.
Many model structures have been proposed for the identification of nonlinear systems. The nonlinear static block followed by a dynamic block in the Hammerstein structure has been found to be a simple and effective representation for capturing the dynamics of typical chemical engineering processes, such as distillation columns and heat exchangers. Nonlinear system identification involves the following tasks: structure selection, including selection of suitable nonlinear model structures and the number of model parameters; input sequence design, including the determination of the input sequence u(t), which is injected into the system to generate the output sequence y(t); noise modeling, which includes the determination of the dynamic model which generates the noise input w(t); parameter estimation, which includes estimation of the remaining model parameters from the dynamic system data u(t) and y(t), and the noise input w(t); and model validation, including the comparison of system data and model predictions for data not used in model development.
In the Hammerstein, a nonlinear system is represented as a nonlinear memoryless subsystem f(.) followed by a linear dynamic part. The input u(t) and the output y(t) are observable, but the intermediate signal v(t) is not. As shown in FIG. 2, the static nonlinear element scales the input u(t) and transforms it to v(t) through a nonlinear arbitrary function ƒ(u). The dynamics of the system are modeled by a linear transfer function, whose output is y(t).
Many different techniques have been proposed for the black-box estimation of Hammerstein systems from input-output measurements. These techniques mainly differ in the way that static nonlinearity is represented and in the type of optimization problem that is finally obtained. In parametric approaches, the static nonlinearity is expressed in a finite number of parameters. Both iterative and non-iterative methods have been used for determination of the parameters of the static-nonlinear and linear-dynamic parts of the model. Typical techniques, however, are extremely costly in terms of computational time and energy. Thus, a method for identifying Hammerstein models solving the aforementioned problems is desired.